Download Notes

Gravitation potential energy
• We need a new formula because Ug = mgh
assumes that h is small as compared to the
radius of the earth and therefore changes in g
were negligible
• But in orbit, there will be variations in g due to
the large distance from the center of the earth
U g  mgh
GmE
U g  mo 2 r
r
Gm1m2
Ug  
r
Why is it negative?
• We choose Ug=0 at r=∞
• We must add energy in order to bring an
object farther from the earth
• This means it will become less negative
Escape Speed
Consider a rocket leaving the earth.
It usually goes up, slows down,
and then returns to earth. There
exists an initial minimum speed
that when reached the rocket
will continue on forever. Let's
use conservation of energy to
analyze this situation!
At infinity, we know that Ug = 0, also
kinetic energy will be 0 because it
will not move anymore at infinity so
therefore the total mechanical
energy will always equal ZERO
Escape Speed
Escape speed vs orbital speed
Escape Velocity =
2GmE
r
Orbital Velocity =
GmE
r
Example
• A rocket is launched vertically from the
surface of the earth with an initial velocity of
10000 m/s. What maximum height does it
reach? (g is not constant)
K Ug  K Ug
1 2
Gm1m2
Gm1m2
mv  
 0
2
r
r
1 2 Gme
Gme
v 

2
re
r
1
6.67 x1011 (6 x1024 )
6.67 x10 11 (6 x1024 )
2
10000 

2
6.37 x106
r
7
r  2.38 x10 m
Gravitational field of hollow shell
• Inside a hollow sphere, the gravitation field is
0. Outside a hollow sphere, you can treat the
sphere as if its entire mass was concentrated
at the center.
Gravitational field of solid sphere
• Outside a solid sphere, treat the sphere as if
all the mass is at the center of the sphere
(same as hollow). Inside the sphere, treat the
sphere as if the mass inside the radius is all at
the center. Only the mass inside the “radius of
interest” counts
Kepler's Laws
There are three laws that Johannes Kepler formulated when he was
studying the heavens
THE LAW OF ORBITS - "All planets move in elliptical orbits, with the Sun at
one focus.”
THE LAW OF AREAS - "A line that connects a planet to the sun sweeps out
equal areas in the plane of the planet's orbit in equal times, that is, the
rate dA/dt at which it sweeps out area A is constant.”
THE LAW OF PERIODS - "The square of the period of any planet is
proportional to the cube of the semi major axis of its orbit."
Kepler’s 1st law – The Law of Orbits
"All planets move in elliptical orbits, with the
Sun at one focus.”
Kepler’s 2nd Law – The Law of Areas
"A line that connects a planet to the sun sweeps out equal
areas in the plane of the planet's orbit in equal times,
that is, the rate dA/dt at which it sweeps out area A is
constant.”
Kepler’s 3rd Law – The Law of Periods
"The square of the period of any planet is proportional to
the cube of the semi major axis of its orbit."
Gravitational forces are centripetal, thus
we can set them equal to each other!
Since we are moving in a circle we can
substitute the appropriate velocity formula!
The expression in the RED circle derived by setting
the centripetal force equal to the gravitational force
is called ORBITAL SPEED.
Using algebra, you can see that everything
in the parenthesis is CONSTANT. Thus the
proportionality holds true!
Kinetic Energy in Orbit
Using our ORBITAL SPEED
derived from K.T.L and the
formula for kinetic energy
we can define the kinetic
energy of an object in a bit
more detail when it is in
orbit around a body.
The question is WHY? Why do we need a new equation for kinetic
energy? Well, the answer is that greatly simplifies the math. If we use
regular kinetic energy along with potential, we will need both the orbital
velocity AND the orbital radius. In this case, we need only the orbital
radius.
Total Energy of an orbiting body
Notice the lack of
velocities in this
expression as mentioned
in the last slide.
So by inspection we see that the kinetic energy function is always
positive, the potential is negative and the total energy function is negative.
In fact the total energy equation is the negative inverse of the kinetic.
The negative is symbolic because it means that the mass “m” is BOUND
to the mass of “M” and can never escape from it. It is called a BINDING
ENERGY.
Energy from a graphical perspective
As the radius of motion gets
larger. The orbiting body’s
kinetic energy must decrease (
slows down) and its potential
energy must increase ( become
less negative).